chapter 12

Crystalline Solids

• Definition: Crystalline solids are materials whose atoms, ions, or molecules are arranged in an orderly repeating lattice.
• Study Method: They are analyzed using X-ray diffraction, a technique where X-rays are scattered from the atomic planes within the solid.
• Governing Principle: These interactions are governed by Bragg's Law.

Bragg's Law

• Principle: When X-rays strike parallel planes of atoms or molecules in a crystal, constructive interference occurs if the difference in the path length between beams reflected from adjacent planes is an integral number of wavelengths.
  • Constructive Interference: This type of interference increases the amplitude of the electromagnetic wave.
  • Destructive Interference: This results in the subtraction of wave amplitudes, effectively canceling them out.
• Memory Aids:
  • For Destructive Interference: Think of destruction meaning things are gone (subtraction).
  • For Constructive Interference: Relate it to building or putting things together.
• Application: Bragg's law helps scientists determine the distance between layers of atoms or molecules in a crystal mass.

Example Problem

• Problem Statement: When an X-ray beam of wavelength 100.154 pm is incident on an iron crystal surface, it produces maximum reflection at an angle of 32.6 degrees. Assuming n=1, calculate the separation between layers of iron atoms in the crystal.
• Solution Steps:
  1. Use Bragg's Law ($n\frac{ ext{lambda}}{2} = d ext{sin}( heta)$) to find the distance between atomic layers.
  2. Rearrange the equation to solve for $d$ (distance).
  3. Plug in values and perform the calculation.
• Final Answer: The separation between layers of iron atoms is 143 pm.

Crystal Lattice

• Definition: The arrangement of particles in a crystalline solid is referred to as a crystal lattice.
• Unit Cell: The smallest unit that fully reflects the arrangement pattern of crystals is called a unit cell.
  • Lattice Points: May consist of atoms, ions, or molecules, while lattice planes connect equivalent points throughout the crystal.
• Characteristics of Unit Cells:
  • Typically three-dimensional.
  • Consist of two or three layers of particles that repeat to form macroscopic crystal structure.
  • Can start from any point within the crystal, leading to the same resulting unit cell arrangement.

Types of Crystalline Lattices

• Chart Overview:
  • Simple Cubic: Each corner of the cube has a lattice point.
  • Body-Centered Cubic (BCC): Contains lattice points at each corner of the cube plus an additional lattice point at the center.
  • Face-Centered Cubic (FCC): Contains lattice points at each corner and the center of each face of the cube.

Coordination Number

• Definition: The coordination number refers to the number of particles each particle in the crystal is in contact with.
• Variation: It varies depending on the type of ions in contact:
  • For ions, it's the number of oppositely charged ions in contact.
• Implication: A higher coordination number indicates more interactions, resulting in stronger attractive forces holding the crystal together.

Packing Efficiency

• Definition: The packing efficiency is the percentage of volume in unit cells that is occupied by particles.
• Cubic Unit Cells: Defined as having equal edges and angles of 90 degrees.
• Calculation: If the unit cell is comprised of spherical particles,
  • One-eighth of each corner particle lies within the cube.
  • One-half of each edge particle does likewise.
  • One-fourth of each face particle resides within the unit cell.
• Application: This leads to calculations of packing efficiency using these fractions based on the type of cells being analyzed.

Body-Centered Cubic Example

• Particle Arrangement: In a BCC unit cell,
  • One-eighth of each corner particle from the eight corners contributes to the unit cell.
  • One complete atom is located in the cube’s center, resulting in two total particles per unit cell.
• Coordination Number: This is calculated to be 8, reflecting the atom's contacts with its surroundings.

Practice Problems

1. Unit Cell Volume and Radius Calculation:

   • A body-centered cubic unit cell has a volume of $4.32 imes 10^{-23} ext{ cm}^3$. Find the radius of the atom.
   • To solve:
     • First, determine the edge length from the volume ($V = l^3$).
     • Use the edge length to calculate the radius of the atom using $l = 2 imes r imes \frac{ ext{sqrt}(3)}{1}$.
     • Convert result from cm to pm.
   • Final result: The radius is determined to be 152 pm.

2. Density Calculation:

   • Aluminum Crystal Example: Aluminum has a face-centered cubic unit cell with a radius of 143 pm.
   • Calculate the density of solid crystalline aluminum in $ext{g/cm}^3$.
   • To solve:
     • Use the packing efficiency and mass of aluminum.
     • Insert proper values to get exact density calculation results.